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Treatise On Those Parts of Geometry Needed by Craftsmen

Sand Geometry
Light Geometry
Egyptian Geometry
Greek Geometry
Roman Geometry
Arabic/Islamic Geometry
Persian Geometry
Pythagoras Geometry
Euclidean Geometry
Vitruvian Geometry
Archimedes Geometry
Apollonios Geometry
Sacred Geometry
Vesica Piscis Geometry
Ad Triangulum Geometry
√3 Geometry
Ad Quadratum Geometry
√2 Geometry
Golden Ratio phi = (√5 + 1) ÷ 2 ) = 1.6180339887

Ad Triangulum ( triangle within the circle)
Ad Triangulum (hexagonal base)
Ad Triangulum (√3 base)
Ad Triangulum (three point geometry)
Ad Triangulum (six point geometry)
Ad Triangulum (human consciousness)
Ad Triangulum (Heaven)
Star of David or Solomon's seal
Thunder Mark's

Ad Quadratum (square within the circle)
Ad Quadratum (octagonal based)
Ad Quadratum (√2 based)
Ad Quadratum (four point geometry)
Ad Quadratum (eight point geometry)
Ad Quadratum (earth geometry)
Ad Quadratum (physical world)

The circle, and its centre, are the point at which all Geometry begins.
From the circle comes three fundamental figures in Geometry, the triangle, square and hexagon.

The ellipse, and its centre, are the point at which all Elliptical Geometry begins.
By plotting abscissas and ordinates of the sun's shadow, you can drive the chariot of the sun's arc south of the equator to see the chariot of light at winter solstice.

Philibert De l'Orme, LE TROISIEME LIVRE DE L’ARCHITECTURE
I will not dwell further on this point, to regain our lines, which are traits of a crossbow to offend, but traits and practices to teach geometry, architecture and the secret worthy of being known, and executed.

Six kinds of lines or geometric figures extracted from Euclid and Archimedes. The first kind will be used for all runs and vaulted cellars strange as they please, as we have said and shown at the beginning of the third book, the other will be used to find all kinds of arches and doors, and the third for all tubes, and the fourth for all kinds of spherical vaults and other developments, the fifth for all ways of stairs, and the sixth for all kinds of screws.

This work contains the science and practice of construction geometry in a simple and familiar manner for the advantages of readers not yet acquainted with geometry or trigonometry, no more trigonometry has been employed than was absolutely necessary. And further, there is always danger of overstocking the average craftmen’s storehouse of knowledge, the result of which tends to confuse and produce errors.

Polygon Rafter Tables
3 Sided Polygon Rafter Table
4 Sided Polygon Rafter Table
5 Sided Polygon Rafter Table
6 Sided Polygon Rafter Table
7 Sided Polygon Rafter Table
8 Sided Polygon Rafter Table
9 Sided Polygon Rafter Table
10 Sided Polygon Rafter Table
11 Sided Polygon Rafter Table
12 Sided Polygon Rafter Table
16 Sided Polygon Rafter Table
24 Sided Polygon Rafter Table
32 Sided Polygon Rafter Table
36 Sided Polygon Rafter Table
42 Sided Polygon Rafter Table
48 Sided Polygon Rafter Table
64 Sided Polygon Rafter Table
96 Sided Polygon Rafter Table

Use of the Framing Square for Octagonal Roofs
Octagon Framing Square usage: 3:12 Pitch
Octagon Framing Square usage: 4:12 Pitch
Octagon Framing Square usage: 5:12 Pitch
Octagon Framing Square usage: 6:12 Pitch
Octagon Framing Square usage: 7:12 Pitch
Octagon Framing Square usage: 8:12 Pitch
Octagon Framing Square usage: 9:12 Pitch
Octagon Framing Square usage: 10:12 Pitch
Octagon Framing Square usage: 11:12 Pitch
Octagon Framing Square usage: 12:12 Pitch

Polygon Rafter Tables all in one PDF file
Polygon Rafter Tables

Eagle Square Manufacturing Co., (South Shaftsbury, Vt.),the Octagon Framing Square that real roofers used !
Eagle Framing Square -- invented in 1814
  
HAWES PAT
Silas Hawes made squares in Shaftsbury, VT, 1814 - 1828, but that several other local makers also marked their squares "HAWES PAT". These were predecessors to the famous Eagle Square Co. organized in 1859. A fine example of a used hand forged, hand stamped square of the early days of the republic.
  
Picture of Eagle Octagon Framing Square
Picture of similar Octagon Framing Square


  

PDF files of geometry construction for Treatise On Those Parts of Geometry Needed by Craftsmen.

Treatise On Those Parts of Geometry Needed by Craftsmen
Treatise On Those Parts of Geometry Needed by Craftsmen
Double Curvature Arch Geometric Development
Intersection of Vertical and Horizontal Cylinders
Double Curvature Arch
Square Groin Vault
With Transverse and Longitudinal ribs Semi-circular
Quadripartite vaulting
with equal height ordinates
Square Groin Vault
Square Groin Vault Segmental Arches
With Transverse and Longitudinal ribs with
segmental arches
Quadripartite vaulting
with equal height ordinates
Square Groin Vault Segmental Arches
Rectangular Segmental Arch Groin Vault
With Transverse and Longitudinal ribs with
segmental arches
Quadripartite vaulting
with equal height ordinates
Rectangular Segmental Arch Vault
Square Gothic Vault with Elliptic Cross Diagonals
With Transverse and Longitudinal ribs
being Gothic Arches
Quadripartite vaulting
with equal height ordinates
Square Gothic Vault with Elliptic Cross Diagonals
Rectangular Gothic Vault
With Transverse and Longitudinal ribs with
Gothic Arches
Quadripartite vaulting
with equal height ordinates
Rectangular Gothic Arch Vault
Octagon vaulting with Equal Side Lengths
With Transverse ribs Semicircular
Octopartite vaulting
with equal height ordinates
Octagon vaulting with Equal Side Lengths Semicircular
Octagon vaulting with Equal Side Lengths
With Transverse ribs Segmental
Octopartite vaulting
with equal height ordinates
Octagon vaulting with Equal Side Lengths Segmental
Octagon vaulting with Equal Side Lengths
With Transverse ribs Gothic
Octopartite vaulting
with equal height ordinates
Octagon vaulting with Equal Side Lengths Gothic
Hexagon vaulting with Equal Side Lengths
With Transverse ribs Lancet Gothic
with equal height ordinates
Hexagon vaulting with Equal Side Lengths Lancet Gothic
Dodecagon vaulting with Equal Side Lengths
With Transverse ribs Lancet Gothic
with equal height ordinates
Dodecagon vaulting with Equal Side Lengths Lancet Gothic
Domical - Spherical Vault on Square base
With Cross Diagonal ribs Semicircular
With Transverse ribs Semicircular
( Sail Vault ) Hemispherical Dome Vault
Domical - Spherical Vault on Square base Semicircular
True Gothic Vault with arc's forming all of the ribs from the book
ROBERT WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES (1842)
Fig. 10 page 13
With Transverse vault ribs, Longitudinal vault ribs stilted, Cross Diagonal ribs, Tierceron ribs are all arc's.
Willis shows how none of the ribs are drawn from ordinates like the Roman and Italian groined vault.
Willis proclaims that Philibert de I'Orme, Maturin Jousse, Derand, De la Rue, Frezier, William Halfpenny and Peter Nicholson were all miss guided by masonic projection techniques. True gothic vaults used only arc's not ellipses from ordinate projection.
ROBERT WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES
Elliptical Surface Development of Barrel Vault Surface
with equal height ordinates
Elliptical Surface Development of Barrel Vault Surface
Surface Development of Cone
Surface Development of Cone
Tangent To Circle
Tangent To Circle
Dihedral Angle of Tetrahedron
Equilateral Triangular Pyramid
Dihedral Angle = 70.52878°

Tetrahedron Angles
D Angle = 30.00000
A Angle = 70.52878
C Angle = 54.73561
E Angle = 30.00000
B Angle = 54.73561

90-D Angle = 60.00000
90-A Angle = 19.47122
90-C Angle = 35.26439
90-E Angle = 60.00000
90-B Angle = 35.26439


Hip Pitch Angle = arctan( tan( Pitch Angle ) * sin( Plan Angle ))
Hip Pitch Angle = arctan( tan( 70.52878° ) * sin( 30° )) = 54.73561°

Hip Backing Angle = arctan( sin( Hip Pitch Angle) ÷ tan( Plan Angle ) )
Hip Backing Angle = arctan( sin( 54.73561°) ÷ tan( 30° ) ) = 54.73561°

Dihedral Angle = ( 90° - Hip Backing Angle ) * 2
Dihedral Angle = ( 90° - 54.73561 ) * 2 = 70.52878°

The dihedral angle is the angle measured between two planes.

Dihedral Angle of Tetrahedron

Dihedral Angle of Rectangular Pyramid Roof
Major Pitch Angle = 39.81°
Minor Pitch Angle = 33.69°

Hip Pitch Angle = arctan( tan( Pitch Angle ) * sin( Plan Angle ))
Hip Pitch Angle = arctan( tan( 39.81° ) * sin( 38.66° )) = 27.50°

Hip Backing Angle = arctan( sin( Hip Pitch Angle) ÷ tan( Plan Angle ) )
Major Pitch Hip Backing Angle = arctan( sin( 27.50°) ÷ tan( 38.66° ) ) = 29.99°
Minor Pitch Hip Backing Angle = arctan( sin( 27.50°) ÷ tan( 51.34° ) ) = 20.27°

Dihedral Angle = 180° - 29.99° -20.27° = 129.74°
The dihedral angle is the angle measured between two planes.

Dihedral Angle of Rectangular Pyramid Roof

Rectangular Pyramid Roof With Square Tail Fascia Geometric & Trigonometric Roof Framing Development
Deck Angle = 90.00000
Plan Angle = 51.34019
Roof Pitch = 8:12
Roof Pitch Angle = 33.69007
R1 Hip Rafter Angle = 27.50055
C5m Hip Rafter Backing Angle = 20.27452
C5m 90° - Hip Rafter Backing Angle = 69.72548
R4m Hip Rafter Side Cut Angle = 35.35971
P2m Jack Rafter Side Cut Angle = 33.64933
90° - P2m Roof Sheathing Angle = 56.35067
P1m Purlin Miter Angle = 23.92974
C1m Purlin Saw Bevel Angle = 31.31734
SFMm Fascia Miter Angle = 34.73648
SFBm Fascia Saw Bevel Angle = 28.67913
R2m Hip Rafter Square Tail Miter Angle = 12.98849
C1m Hip Rafter Square Tail Saw Bevel Angle = 31.31734

Deck Angle = 90.00000
Plan Angle = 38.65981
Roof Pitch = 10:12
Roof Pitch Angle = 39.80557
R1 Hip Rafter Angle = 27.50055
C5a Hip Rafter Backing Angle = 29.99339
R4a Hip Rafter Side Cut Angle = 47.95238
P2a Jack Rafter Side Cut Angle = 43.83911
90° - P2a Roof Sheathing Angle = 46.16089
P1a Purlin Miter Angle = 38.66786
C1a Purlin Saw Bevel Angle = 36.86131
SFMa Fascia Miter Angle = 27.11914
SFBa Fascia Saw Bevel Angle = 40.52065
R2a Hip Rafter Square Tail Miter Angle = 25.64298
C1a Hip Rafter Square Tail Saw Bevel Angle = 36.86131

Rectangular Pyramid Roof With Square Tail Fascia Geometric & Trigonometric Roof Framing Development

Treatise on Stair Building & Handrailing
by W & A Mowat
page 178, fig.214
Twist Bevel Angle Geometric Development based on Tetrahedron of Triangular Pyramid
Dihedral Angle = mPn
Twist Bevel Angle = bVc
Bevel Angle on joint line tangent = aV1c
D A C E B Tetrahedron Angles
D S R1 P2 C5 Hawkindale Angles

Twist Bevel Angle Geometric Development based on Tetrahedron of Triangular Pyramid

Helix Geometric Development
Circumference = pi x Diameter
Length of Helix = sqrt (pi x Number of Turns x Diameter)^2 + (Length^2)
Helix Geometric Development
Dome on Octagonal Pyramidal Roof using equal height ordinates for geometric development.
Dome on Octagonal Pyramidal Roof Geometry
Ellipse Geometric Development with Radical Lines .
Ellipse Geometric Development with Radical Lines
Cone Surface Geometric Development.
Cone Surface Geometric Development

Barrel Vault Intersects Main Roof Vertical & Horizontal Trace Development.

If a line be perpendicular to an oblique plane, the projections of the line are
respectively perpendicular to the traces of the plane; that is, the horizontal
projection to the horizontal trace, and the vertical projection to the vertical trace.

Barrel Vault Intersects Main Roof Vertical & Horizontal Trace Development

Equal Sided Octagon Geometric & Trigonometric Roof Framing Development
Pitch Angle = 33.69°
Plan Angle = 67.5°

Hip Rafter Pitch Angle = arctan( tan( Pitch Angle ) * sin( Plan Angle )) = 31.63°
Hip Rafter Backing Angle = arctan( sin( Hip Rafter Pitch Angle) ÷ tan( Plan Angle ) ) = 12.26°
Hip Rafter Side Cut Angle = arctan( cos( Hip Rafter Pitch Angle ) ÷ tan( Plan Angle )) = 19.43°

Jack Rafter Side Cut = arctan( cos( Pitch Angle ) ÷ tan( Plan Angle )) = 19.01°
Roof Sheathing Angle = arccos( cos( Plan Angle ) * cos( Hip Rafter Pitch Angle )) = 70.98°

Roof Sheathing Cut Measurement = Plywood Width ÷ tan( Roof Sheathing Angle) = 16.54"
Jack Rafter Difference = tan( Roof Sheathing Angle ) * Jack Rafter Spacing = 69.62"

Frieze Block Saw Miter Angle = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle )) = 12.94°
Frieze Block Saw Blade Bevel Angle = arctan( sin( Frieze Block Saw Miter Angle ) ÷ tan( Pitch Angle )) 18.57°

Common Rafter Rise = Run * tan( Pitch Angle )
Common Rafter Length = Run ÷ cos( Pitch Angle )
Common JackRafter Run = Jack Rafter Spacing * tan( Plan Angle )
Common JackRafter Length = Jack Rafter Runn ÷ cos( Pitch Angle )

Hip Rafter Run = Run ÷ cos( Plan Angle )
Hip Rafter Length = Hip Rafter Run ÷ cos( Hip Rafter Pitch Angle )


Equal Sided Octagon Geometric & Trigonometric Roof Framing Development

Polygon 2 Cord Gables ( Prow Rafters ) can be calculated with the same angles used to calculate the polygon rafters. The working angle of the polygon is used to calculate the length of the 2 Cord Gable. The length of the 2 Cord Gable is calculated using the following formulas.

Example:
Polygon = Octagon
pitch = 8:12
pitch angle = 33.69°

side wall length = 59.64696
hip rafter run = side wall length
common rafter run = hip rafter run × cos ( working angle )
common rafter run = 59.64696 ÷ cos ( 22.5° ) = 52.33496
roof rise = common rafter run × tan ( pitch angle )
roof rise = 52.33496 × tan ( 33.69° ) = 34.88988hip pitch angle = arctan ( tan ( pitch angle ) × sin ( plan angle ) )
hip pitch angle = arctan ( tan ( 33.69° ) × sin ( 67.5° ) ) = 31.63°

common rafter length = common rafter run ÷ cos ( pitch angle )
common rafter length = 52.33496 ÷ cos ( 33.69° ) = 62.89874

hip rafter length = hip rafter run ÷ cos ( hip pitch angle )
hip rafter length = 59.64696 ÷ cos ( 31.63° ) = 70.05

polygon gable rafter length = hip rafter length

Mark the gable rafter at the same pitch angle as the hip pitch angle on the gable rafter material. Then cut the head cut angle with your saw set at the polygon miter angle. The Octagon polygon miter angle is 22.5°. The saw bevel angle is the same as the polygon miter angle.

The polygon 2 cord gable rafter (prow rafter) is calculated at the same pitch angle as a polygon hip rafter.

Polygon 2 cord gable rafter (prow rafter) Geometric & Trigonometric Roof Framing Development
Polygon 2 Cord Gable Crown Molding Angles and Prow Rafter Development Calculators

Tetrahedron Developmental Geometric & Trigonometric Developmental drawings for Crown Molding with 38° Spring Angle .
Crown Molding with 38° Spring Angle Geometry
Tetrahedron Developmental Geometric & Trigonometric Developmental drawings for 8:12 pitched roof.
Tetrahedron Developmental Geometric & Trigonometric Developmental drawings for 8:12 pitched roof
Canadian & American Geometric Roof Framing Development & Framing Square Usage.
Canadian & American Geometric Roof Framing Development & Framing Square Usage Tetrahedron Calculator
Canadian & American Geometric Roof Framing Development & Framing Square Usage
Irregular hip roof Geometric & Trigonometric Developmental drawings for 6:12 - 12:12 pitched roof & Framing Square Usage.
Irregular hip roof Geometric & Trigonometric Developmental drawings for 6:12 - 12:12 pitched roof
Roof Framing Kernel drawing for visualization posterboard cut out model .
Roof Framing Kernel drawing for visualization posterboard cut out model

Ellipse Formulas for Eyebrow Roof Design

Intersecting Slope Angle = arctan((Roof Pitch - Dormer Pitch ) ÷ 12)
Semi Minor Axis = Radius
Roof Surface Semi Major Axis = ( Radius ÷ tan( Intersecting Slope Angle ) ) ÷ cos( Roof Pitch Angle )
Plan View Semi Major Axis = Radius ÷ tan( Intersecting Slope Angle )
Elevation View Semi Major Axis = ( Radius ÷ tan( Intersecting Slope Angle ) ) * tan( Roof Pitch Angle )


Ellipse Formulas for Eyebrow Roof Design

Ad Triangulum based on three point geometry, Medieval Cathedral Framing based on Ad Trianguluml Geometry.
Ad Triangulum 3 point Geometry
Ad Triangulum based on six point geometry, Medieval Cathedral Framing based on Ad Trianguluml Geometry.
Ad Triangulum 6 point Geometry
Vitruvius - Ptolemy Alalemma Sundail from 12 point geometry and orthographic projection of ellipse.
Elliptical Sundial Geometry
Vitruvius The Alalemma Sundail and Its Applications Book 8,Chapter 7
Vitruvius Alalemma Sundail Geometry for Latitude 37.91°
Construction Of Gothic Window with Oculus Based on Euclid's Elements Book 1 Prop 1
Gothic Window with Oculus
Equilateral Gothic Arch Construction Based Euclid's Elements Book 1 Prop 1
Equilateral Gothic Arch
Six Point Geometry, Euclidean Geometry, Seed Of Life, Hex Rose Pedal,Daisy Wheel, Thunder Marks, Construction Based Euclid's Elements Book 1 Prop 1
Six Point Geometry, Euclidean Geometry, Seed Of Life, Hex Rose Pedal,Daisy Wheel, Thunder Marks
Square Construction Based Euclid's Elements Book 1 Prop 1
Square Construction
Pentagon Construction Based Euclid's Elements Book 1 Prop 1
Pentagon Construction
Equilateral Arch Construction Based Euclid's Elements Book 1 Prop 1
Equilateral Arch
Elliptic Gothic Cross Vault Arch Construction Based Euclid's Elements Book 1 Prop 1
Elliptic Gothic Cross Vault Arch
Euclid's Elements Book 1 Prop 47 In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.
Euclid's Elements Book 1 Prop 47
Philibert De l'Orme, LES LIVRES D'ARCHITECTURE
Squaring the Foundation with Equilateral Triangles

Take the case that you have drawn the line QR, and hereunto an equilateral triangle, that is to say as great on one side than the other, as you see RST, T is the point where you pull another curved line marked Z, is tightened without moving the compass, and requires that the distance ST is similar to that of TZ. This makes you draw a straight line from point S to T, till it intersects the line Z, and this place, as you see the point marked X, you draw another line even to the point of R, which will be precisely perpendicular to the line QR, as you can judge by the figure followed.


Philibert De l'Orme, LES LIVRES D'ARCHITECTURE, Squaring the Foundation with Equilateral Triangles Geometry
The flat plane surfaces of the Platonic Solids are the triangle, square, and pentagon.
Platonic Equilateral Triangle, Square, Pentagon Geometry
Pyriamd Geometry with phi Golden Ratio phi = (√5 + 1) ÷ 2 ) = 1.6180339887
Pyriamd Geometry with phi
Vitruvius Greek Theater based on Square,Ad Quadratum Geometry,Book 5,Chapter 7
Vitruvius Greek Theater based Ad Quadratum Geometry
Philibert De l'Orme, LES LIVRES D'ARCHITECTURE
Wind Direction of the four corners of the world.
Rotated Squares
Ad Quadratum Geometry
Chapter 7

Of these four main parts of the world four winds blowing directly appointed principal or cardinal knowledge is the point of East Subsolanus, marked the ensuing Figure A, where the quality and nature is hot and dry in the West Favonius sale, noted by C, its quality being cold and moist, Midi, Auster, signed B, whose nature and quality wet and hot and Septentrion Boreas, marked D, where quality is cold and dry.
So much for four parts and corners of the world with their own domestic wind.
It should therefore be noted that the ancients still divided equally into four one each space between the aforesaid principal winds, and gave one each to own a wind superabundant. Whereby between Subsolanus and Auster, that is to say between East and South, or if you want between A and B also, they have located the wind called Eurus, marked by E, between South and West, Africus noted by F . , Between West and north, signed by G. Caurus, and between East and Septentrion Aquilo, marked by H.


Philibert De l'Orme, LES LIVRES D'ARCHITECTURE, Ad Quadratum 8 point Geometry
Philibert De l'Orme, LES LIVRES D'ARCHITECTURE, Ad Quadratum 16 point Geometry
Vitruvius Roman Theater based Triangle, Ad Triangulum Geometry, Book 5,Chapter 6 Chapter 5
1. The foundation should be of the soundest workmanship and materials, and of greater thickness than the walls above.
2.The plan of a city should not be square, nor formed with acute angles, but polygonal.
3. The thickness of the walls should be sufficient for two armed men to pass each other with ease.
4. The distance between each tower should not exceed an arrow's flight.
5. The towers should be made either round or polygonal.

Vitruvius Roman Theater based Triangle, Ad Triangulum Geometry

Daisy Wheel based on six point geometry, Timber Framing based on Daisy Wheel Geometry
Timber Framing based on Daisy Wheel Geometry
Golden Ratio phi Geometry, Two Story Framing based on Golden Ratio, Golden Ratio Geometry based on equilateral triangle inscribed in circle,Golden Ratio Geometry based on pentagon inscribed in circle
Golden Roof Slope Angle 31.72°
Gothic Arch Semi-Major Radius for a 67.5° Plan Angle, 60° Plan Angle, 45° Plan Angle, 30° Plan Angle Geometry
Gothic Arch Elliptical Cross Vault Hip Drawn with Height Ordinates Geometry
Gothic Arch Types Geometry,Standard Gothic Arch with Radius equal to width of opening, Lancet Gothic Arch with Radius bisecting radius cord and spring line of opening
Lancet Gothic Arch Types
Octagon Rosette Geometry
Octagon Rosette
Descriptive Geometry for Drawing an Octagon, Sacred Cut
Octagon Geometry
Descriptive Geometry for Drawing an Octagon from side length of Octagon
Octagon from side length of Octagon Geometry
Descriptive Geometry for Octagram
Octagram Geometry
Bisect Angle using a Compass and Straight Edge
Bisect Angle using a Compass and Straight Edge
Descriptive Geometry, How To Draw Oval for Elliptical Cross Vaults Using only Compass and Straight Edge
How To Draw Oval for Elliptical Cross Vaults Using only Compass and Straight Edge
Gothic Cross Vault, Elliptical Cross vault diagonals drawn with co-centric circles and orinadtes.
co-centric circles and orinadtes
Gothic Arch from vesica piscis
Gothic Arch from vesica piscis
Gambrel Roof Design Geometry from Dutch Hexagram - Rotated Hexagons
Gambrel Roof Design
Philibert De l'Orme, LES LIVRES D'ARCHITECTURE
Speed Square or Angle Dial with an Equilateral Triangle
Ad Triangulum Geometry

Be given an equilateral triangle such width as you like, like ABC, the more it grows, so will the insurance and kindness. Where do I did not wish to help more than that which you see below figuratively, by as much as I used to be easier in my coffers, and do not usually point thereof, a Astrolable be, and ephemeris, with a few other books, and cases filled compass, and what it takes to portraire. Within this triangle imagine a circle, as you can see marked EFGH (almost as if it were a dial showing the hours) and divide into so many parts that will, like twenty-four, thirty-two, forty eight, the most that there is the best. I divided the latter into thirty-two, and is set amidst a magnetic needle, as well as marine dials and compasses, or small whom we help to find the hours to the Sun, but notice that said needle must be very good and very moving. When you want to help the triangle, you look through one side as you please, for the one marked in Figure D. This makes you discard your city view, castle or place from which you want to take the form and figure, and make a sketch first on paper scored coarsely and you can understand the decision. Can you make the trip at all. If you want that it should fit in memory or writing a bend and each face of the walls to measure the length as you will see below. Having done this, you can start at one end of the castle, town or place, keeping your triangle against the first section of wall with a ruler to have greater decision, against which must be your triangle as you see marked K.


Philibert De l'Orme, LES LIVRES D'ARCHITECTURE, Angle Dial with an Equilateral Triangle Geometry
Philibert De l'Orme, LES LIVRES D'ARCHITECTURE
Plumb Bob from an Equilateral Triangle
Ad Triangulum Geometry

Philibert De l'Orme, LES LIVRES D'ARCHITECTURE, Plumb Bob from Equilateral Triangle Geometry

Philibert De l'Orme, LE TROISIEME LIVRE DE L’ARCHITECTURE
The three lost points of the circle geometry and lengthened arch

I suggest you put three points to your will, and that from one point to another you draw the lines, you divide that by the middle, and then made a perpendicular thereon, and you see the two lines A and B and where they meet and intersect is the center and you see the place where is C, where you have to put one of the points of the compass, and mark another line precisely, which will go on three points, as you can see in figure marked C in the center.

You can also proceed in this case with the compass by the way you see kept in the following figure given, which is way more secure. So that those who are quick to wield said compass, do not square, as well as if it is just and right, the line can not be done precisely. So looking to find the lengthened with the three points is very useful and necessary, because you can not only do not lift a panel to a building on a round shape, it must always find you are looking lengthened, which can not be done quickly if not by those three points lost, they are in the panel as those marked D and I said, and are looking more and different.


Philibert De l'Orme, LES LIVRES D'ARCHITECTURE, The three lost points of the circle geometry and lengthened arch
Philibert De l'Orme, LE TROISIEME LIVRE DE L’ARCHITECTURE
Geometric features that show as it is cut and cut the stones to make the doors and down the basement and floors that are in the land, such as kitchens, baths, bathe, and similar where you can go to level, and it must descend.

Philibert De l'Orme, LES LIVRES D'ARCHITECTURE, Sloped basement cellar Vault geometry
Philibert De l'Orme, LE TROISIEME LIVRE DE L’ARCHITECTURE
Modern vaults, the master masons were accustomed to churches and homes of the nobles.

Philibert De l'Orme, LES LIVRES D'ARCHITECTURE, Modern Cross Vault geometry

 

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